Utility-Oriented Precoder Design

• Utility-oriented precoder design is a framework for creating precoding matrices that optimize system-level metrics in MIMO and multi-user networks.
• It integrates objectives such as sum rate, fairness, and power efficiency while addressing practical constraints like PAPR and hardware limitations.
• The approach leverages manifold optimization and structured algorithms to balance performance, complexity, and real-time adaptability.

Utility-oriented precoder construction refers to the principled design of precoding matrices or codebooks for MIMO and multi-user networks to optimize a system-level "utility" objective. Utilities range from sum spectral efficiency and max-min user rate to proportional fairness, power efficiency, minimum pairwise distance for error, peak-to-average power ratio (PAPR), or more general performance measures subject to practical constraints. This paradigm integrates metrics such as achievable rate, power or complexity cost, and implementation constraints into the precoder design, and admits both closed-form and algorithmic solutions depending on the network model, utility, and constraint structure.

1. Utility Formulation: Multi-Criterion and Application-Specific Objectives

Utility-oriented precoder construction starts with an explicit performance metric to optimize, incorporating system model, channel knowledge, and deployment constraints.

Typical utility objectives include:

• Weighted sum-rate maximization: $\sum_{k} w_k R_k$, for user rates $R_k$, under sum or per-user power.
• Fairness metrics: Max-min rate $\max_{P}\min_k R_k(P)$, proportional fair (sum-log utility) $\sum_{k} \log R_k$, or harmonic mean.
• Power efficiency: Incorporation of total, per-user, or per-antenna power, or objectives such as transmit energy per bit or PAPR.
• Robustness and complexity: Minimum pairwise distance for finite constellations (Srinath et al., 2011), average mutual information loss for quantized codebooks (Bhogi et al., 2021), or sparse-structure-induced complexity (Asano et al., 3 Mar 2026).

In advanced settings (e.g. SWIPT-enabled cognitive radio), utility may be a joint function of information throughput, harvested energy, and interference levels (Song et al., 2018).

The composite utility often appears as a weighted sum or constrained optimization, e.g.

$$\max_W \;\;\alpha\, \mathbb{E}_H[R(W)] - \beta\, \mathbb{E}[\mathrm{PAPR}(W)] - \gamma (\mathrm{complexity}(W) + \mathrm{storage}(W))$$

but is typically instantiated via constraints for hard limits (e.g., per-antenna power, PAPR, interference hygiene).

2. Frameworks and Manifold Representations

Modern approaches recognize that the space of feasible precoders is fundamentally manifold-structured, stemming from the invariances and constraints of wireless channels and hardware:

• Grassmannian/Schubert Cell Structure: Beamforming or precoding vectors for unitary, finite codebooks can be seen as points on a Grassmannian $G(T,M)$, often further structured using Schubert cell decompositions to impose sparsity, orthogonality, or other symmetry (Asano et al., 3 Mar 2026).
• Matrix Manifold Optimization: General power, per-user, or per-antenna constraints shape the feasible set as spheres or oblique manifolds in $\mathbb{C}^{M \times N}$, enabling Riemannian gradient and trust-region approaches for non-convex utilities (Sun et al., 2023).
• Tensor Product Structures: For FD-MIMO and UPA arrays, channel and precoder structures admit Kronecker (tensor) factorizations, reducing complexity and allowing codebook design as clustering on product Grassmannians (Bhogi et al., 2021).

By expressing constraints and objectives in this geometric setting, utility maximization is transformed into structured optimization on these manifolds, enabling efficient algorithms with provable convergence for a broad class of systems.

3. Algorithmic Methodologies for Utility Maximization

Algorithmic design for utility-oriented precoding is shaped by the utility function and system constraints. Dominant paradigms include:

a) Convex and Linear-Fractional Reformulations

For monotonic, unitary-invariant utilities (e.g., mutual information, MMSE), joint pilot-precoder design is achieved by decoupling the optimization into alternating subproblems, leveraging the linear-fractional structure of the effective SNR (Pastore et al., 2012). Each subproblem is convex/quasi-convex after appropriate eigen-alignment.

Example:

• Alternating projection: Optimize precoder $Q$ for fixed pilot $P$ (convex), then pilot for fixed $Q$ (convex), project onto Pareto optimal boundary, iterate to convergence (Pastore et al., 2012).
• Closed-form eigenvalue parameterization for Pareto-efficient throughput under per-antenna power (Petrov et al., 13 Aug 2025).

b) Riemannian Optimization on Matrix Manifolds

For high-dimensional and per-antenna/user power-constrained massive MIMO, the constraint set becomes a manifold, and Riemannian (projected) gradient, conjugate gradient (RCG), and trust-region (RTR) schemes are employed (Sun et al., 2023). Key ingredients are:

• Computation of ambient Euclidean gradient of WSR or other utility;
• Tangency projection for the specific power constraint (sphere, oblique, etc.);
• Retraction and vector transport for update feasibility and convergence.

These approaches avoid costly large matrix inverses and achieve fast, scalable convergence in practice.

c) Path-Following and Surrogate-Based Algorithms

For non-convex, nonsmooth models (e.g. multicell NOMA, multicast), optimization is performed via iterative surrogate construction:

• Sequential convex quadratic programs built via concave lower bounds of log-det or min log-det utilities, enforcing surrogates’ tangent and under-estimator properties (Nguyen et al., 2017).
• Monotonic objective improvement and guarantee of stationarity.

d) Submodular Maximization and Clustering Methods

For codebook quantization or discrete codeword selection:

• Product codebook optimization maps mutual information loss to a K-means clustering on the Cartesian product of Grassmannians, using chordal distance as distortion (Bhogi et al., 2021).
• For multicasting, robust submodular maximization under knapsack constraints via greedy or cover algorithms guarantees bicriteria performance for minimum rate or sum-delay (Zhu et al., 2011).

e) Dual/CCCP and Hamiltonian Approaches

For highly-constrained multi-resource problems (e.g. SWIPT-MuMIMO-CR), dualization with ellipsoid subgradient or CCCP (concave-convex) methods enables tractable optimization when direct WMMSE or primal approaches fail (Li et al., 2020, Song et al., 2018, Lin et al., 31 Jul 2025).

• CCCP transforms difference-of-convex objectives/constraints (arising naturally for general utilities) into convergent successive convex approximations per outer iteration (Li et al., 2020).
• Symplectic optimization treats the WSR as a “potential energy” and solves the constrained problem as a dissipative Hamiltonian system, discretized for stability and efficiency (Lin et al., 31 Jul 2025).

4. Structural and Complexity Trade-offs

Optimality is often unattainable at realistic system scales, motivating structured (sparse, block, manifold) designs:

• Sparsity and Codebook Conciseness: Schubert cells and sparsity imposition on the Grassmann manifold yield codebooks with minimal PAPR, storage, and multiplication complexity, maintaining maximal chordal distance packing and nearly optimal achievable rates (Asano et al., 3 Mar 2026).
• Per-antenna and hardware constraints: Efficient parameterizations enable direct navigation of the Pareto front under per-antenna limits, matching the complexity of unconstrained ZF up to a constant, and outperforming when noise is non-negligible (Petrov et al., 13 Aug 2025).
• Product codebooks: Decomposition into low-dimensional K-means problems on component Grassmannians reduces computational burden and storage quadratically compared to full-vector quantization (Bhogi et al., 2021).
• Riemannian methods: Avoidance of repeated large matrix inversions, with convergence to stationary utility-maximizers in tens of iterations, and significant speedups over classical WMMSE (Sun et al., 2023).

These structural insights enable real-time or near real-time adaptation in deployment, especially in massive-MIMO and beyond-5G/6G scenarios.

5. Representative System Results and Performance Insights

System-level evaluations validate the utility-oriented approach:

• Achievable rate and fairness: Sparse codebooks on the Grassmannian outperform 5G NR default codebooks in SNR by $R_k$0 dB while greatly reducing PAPR (from $R_k$1 to $R_k$2 dB at $R_k$3 CCDF threshold) in DFT-s-OFDM uplink (Asano et al., 3 Mar 2026).
• Complexity: Symplectic WSR optimization in UCN-massive MIMO reaches nearly $R_k$4 sum-rate gain over WMMSE with far lower iteration cost (Lin et al., 31 Jul 2025).
• Robustness: The WMMSE-dual method achieves strong duality and global optima even in non-convex SWIPT-CR (Song et al., 2018).
• Multicast: CAA-based algorithms for instantaneous maximum-minimum rate or weighted sum-delay design demonstrate consistent improvement over recursive, open-loop, or simple greedy baselines, with proven convergence/approximation properties (Zhu et al., 2011).

6. Design Recommendations and Best Practices

State-of-the-art research consistently supports:

• Utility tailoring: Select a utility aligned with system-level KPIs (throughput, fairness, power, complexity) and hardware constraints.
• Manifold-based or structure-exploiting algorithms: Employ Grassmannian, tensor, or manifold optimization as these admit strong theoretical properties and computational gains.
• Layered/Alternating or dual approaches: Decompose non-convex or high-dimensional problems to alternating tractable subproblems or employ dual/subgradient/CCCP strategies as needed.
• Sparsity-aware codebooks: Leverage sparsity (via Schubert cells, product constructions, or matching patterns) to minimize PAPR and complexity, especially in uplink/massive-MIMO where PA and RF-efficiency are critical.

By adhering to a utility-oriented, structure-exploiting design methodology, practitioners and researchers can systematically engineer precoding solutions that trade off the critical dimensions of rate, power, fairness, complexity, and robustness in high-dimensional and heterogeneous wireless environments (Asano et al., 3 Mar 2026, Sun et al., 2023, Bhogi et al., 2021, Petrov et al., 13 Aug 2025, Lin et al., 31 Jul 2025, Song et al., 2018, Zhu et al., 2011).